On an aggregation method for CPU-disk models
Citation for published version (APA):Wijbrands, R. J. (1986). On an aggregation method for CPU-disk models. (Memorandum COSOR; Vol. 8605). Technische Universiteit Eindhoven.
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Memorandum COSOR 86-05
On an Aggregation Method for
CPU-Disk Models
by
Rudo J. Wijbrands
ABSTRACT
In this paper an approximation method is discussed for solving some special
types of CPU-disk models by use of aggregation. The basic model is a closed queuing network model of a computer system. The model consists of several CPU's and disk units. Several types of clients are distinguished. The routing matrix for each type of client, however, restricts the clients to visit only one of the CPU·s. The solution method discussed here, is based on the mean value analysis. Numerical experiments are used to show the accuracy of the proposed approximation method. Two types of examples are used here: the aggregation directly applied to the sketched computer model. and used in combination with the parametric analysis of a computer system in a terminal environment.
1. Introduction
One of the most popular algorithms for solving product form closed queuing networks is the mean value analysis (MVA) algorithm (refer Reiser & Lavenberg (15]). It has gained its great popularity thanks to its intuitively appealing formulation; a set of recursive rela-tions entirely in terms of throughputs. sojourn times and queue lengths.
A disadvantage of the algorithm is the computational complexity. For a closed queuing network system with M single server stations and R types of clients to be distinguished. this yields an order of complexity of:
R
OCMRIICK,+l))
(0
,=1
where K, denotes the number of clients of type r in the system. However. up to now, no algorithms have been developed which reach a major break through in this matter. This ahhough it should be noted. that recent developments such as the RECAL algorithm by Conway and Georganas [8]. or the tree MVA algorithm by Tucci and Sauer [19]. for some types of problems do offer some benefit as far as the computational complexity is con-cerned.
In many situations the computational complexity becomes a serious problem. It is there-fore no wonder that many attempts have been made to develop fast approximation methods for solving the performance characteristics of product form networks. Looking at the complexity formula (1) of the MV A-algorithm. one of the attractive ways to reduce the computational complexity of the problem seems to be the reduction of the number of types of clients distinguished. A natural way to do this is by returning to the problem of workload characterization. Usually. a workload is defined by clustering an enormous set of data about the workload. Possibly the types of clients defined by this analysis are quite arbitrary. Allowing the clustering algorithm to work with larger deviations will yield a workload description with less types of clients (refer e.g. Anderberg [2]).
This reduction is less evident if already was chosen for a more logical workload characteri-zation in terms of queues. As an example we mention the modeling of computer systems. A logical workload description may distinguish between jobs according to the arrival pat-terns; arrival patterns which often correspond with the use of certain applications. A com-mon distinction made in models is between batch jobs. per definition entering the system via a(n infinite source) queue. transaction jobs. representing the load of non intensively used on line applications such as a library information ~ystem. and interactive or terminal jobs. representing the "ordinary" users.
*) The investigations were supported in part by the Foundation for Computing Science in the Netherlands. SION. with financial aid from the Netherlands Organization for the Advancement of Pure Research (ZWO).
If we obtained a logical workload description distinguishing several types of jobs. a solu-tion could be to aggregate over all but a single type of client and thus reducing the prob-lem to a relative simple two type model. Next. this model can be used to evaluate the per-formance characteristics for the non aggregated type of client. Defining for each of the types of clients to be studied a separate model yields a set of two type models. which can be used to evaluate the performance characteristics for all types of clients. The computa-tional complexity of this approach is not necessarily less than the complexity of the origi-nal problem. however.
A second way to reduce the computational complexity (1) is to reduce the number of sta-tions M by aggregation. A technique which exploits this option is the parametric analysis introduced by Chandy. Herzog and Woo. [5] and [6]. The parametric analysis first studies a small part of the network as a stand alone system which is not affected by the remaining parts of the network. Next a queue dependent station is defined with the same perfor-mance properties as the part of the system just studied. The latter part of the system is lifted out of the original problem and replaced by this queue dependent server. A relative small problem remains. This technique becomes attractive. if one wants to vary the work-loads in the remaining part of the problem. A fruitful area of application for this tech-nique is the computer-terminal model which is central in this paper. There. the computer system is first studied as a stand alone system. after which on the aggregated level we can study the influence of the number of terminals on the system performance.
In Kritzinger. van Wyk & Krzesinski [12] it is shown that this parametric analysis yields exact results in case we study a product form network as described in Baskett. Chandy. Muntz & Palacios [4].
The extension of this technique to the case where we aggregate several parts of the system. is a trivial one (refer Akyildiz. Bolch & Paterok [1
D.
A disadvantage of the parametric analysis is that on the aggregated level the opportunities for the mean value analysis are rather small. Usually. more complex and less intuitively appealing solution techniques have to be applied.
A third technique which can be used to reduce the complexity (1) of the problem is rooted in the mean value scheme itself. The mean value scheme is based on a recursion which cou-ples the performance characteristics of a system with population vector
1£
to the charac-teristics of all the systems containing a population1£
-£, . which is one client of type r less than popUlation1£.
As the empty system is the only system of which we exactly know all performance characteristics. the recursion has to go back to that leveL A solution which was suggested even before the mean value algorithm had reached it great popularity came from Schweitzer [17. 18]. In terms of the mean value algorithm. he suggested to give an approximate solution for the system with population vector K -~ . one client less then the popUlation vector K we are interested in. In this way one can avoid most of the recursion steps. He suggested to refine the initial solution in an iterative way.The method appears to converge quickly. and therefore. despite the fact that it is not always as accurate as one would wish. has become very popular.
Several improvements to the Schweitzer method have been suggested. The most· popular improvement is the linearizer approximation, introduced in Chandy & Neuse [7]. This algorithm starts the Schweitzer algorithm to approximate K-!!.,: from K-£,.-~ .. and next applies the remaining steps in the mean value scheme to compute the performance charac-teristics for population vector K. Next the additional information gained is used to make a
more sophisticated guess for the performance characteristics which initialized the recur-sion. This initialization can be iteratively improved.
Even without the more sophisticated reinitialization, the linearizer method shows an improved accuracy as compared to the Schweitzer algorithm. This first order depth adjust-ment of the Schweitzer algorithm seems to profit from an apparent ability of the mean value scheme to restore from errors which occurred in the computation of former steps of the recursion. This remarkable observation is studied e.g. in Wijbrands [21].
An approximation method which is also based on this observation, but in addition is able to give bounds on the solution. is the performance bound hierarchy (PBH) algorithm intro-duced by Eager and Sevcik [10]. The original version of this algorithm can be used for sin-gle class closed queuing networks only. It is based on the ordinary mean value scheme, but it initializes half-way the scheme using a rather trivial upper or lower bound. The final
bounds obtained become closer the more recursion steps are used. (
Recently. Eager and Sevdk published a new variant of the PBH algorithm.
[ttl.
This algo-rithm is capable of dealing with multiple closed queuing networks as well. In order to obtain bounds, however. the mean value relations used had to be slightly adjusted.A disadvantage of these iterative algorithms is that the performance characteristics of just a single population are evaluated. Sometimes it might be important to gain information for more populations. One such method is discussed in Van Doremalen [9]. A recursive aggregation technique is introduced which is based on the mean value scheme. Van Doremalen aggregated over a11 types of clients visiting the same service stations, thus reducing the complexity of the mean value algorithm. In each recursion step of the mean value scheme. however. a disaggregation was performed. relating the performance charac-teristics for each of the original types of clients to the performance characcharac-teristics of the system containing a corresponding client of the aggregated type less.
Again. the method can be improved using a first order depth adjustment. This improve-ment is at the price of loosing the advantage of obtaining the performance characteristics for more populations.
The aggregation ideas discussed below are closely related to this technique.
In this paper we study a very special type of problem. We consider a computer system
with a number of processors. Once a cHent in this system has been assigned to a certain processor. it can never receive service from any of the other processors. This situation might occur in a system with a number of tightly coupled processors. sharing both main memory and background memory. whereas each of the processors does have dedicated channels which connect the processor with the background memory (see Figure 1a). A second example is a system with loosely coupled processors. which share background memory. but have private main memory and channels (Figure 1b). As a consequence of this organization. the model we have to construct for this system distinguishes. given the differences in the routing. a number of types of clients which is at least equal to the number of processors in the system. Looking at a more global level. however. the service requirements for the types of clients which thus have to be distinguished. might be the same. regardless of whether the job is assigned to processor i or processor j. This observa-tion suggests a natural way of aggregating; jobs which appear to be identical if they could run on the same processor are to be aggregated. It is the aim of this paper to investigate the possibilities of such an aggregation. Such an aggregation would greatly improve the
tractability of the parametric analysis of a computer-terminal mode1. in which the com-puter is the queue dependent server.
I
CPU
C H A N N E LMAIN
MEMORY
I
CPU
C H A N N E lBACKGROUND
MEMORY
10} Tightly coupkd processors
MAIN
MAIN
MEMORY
MEMORY
I
J
CPU
CPU
c
c
H H A A N N N N E E l lBACKGROUND
MEMORY
1b} Loosely c()upkd processors Figure 1: Exompks of multi processor computer systems
In Section 2 we briefly summarize the mean value scheme on which we shall base our investigations. In Section 3 we discuss three aggregation ideas. In Section 4 we present some numerical results to evaluate these ideas. In Section 5 the computer system under study is embedded in a computer-terminal problem. using the ideas of the parametric analysis. In Section 6. fmally. we add some concluding remarks.
2. The mean value algorithm
To introduce our notations. we shall briefly discuss the mean value algorithm for closed single server station queuing networks. Starting point is the queuing network representa-tion of a computer system. depicted in Figure 2.
We will use the following notation:
r index denoting the type of client. r
=
1.2 ... R system population in vector notation.~
E1
(O.O ... O).(O.O •....• O.Il.. ... (K ,.K , ... .1(.)I
.!t.,. vector denoting a population of a single client of type r
S; t
Lk.]
mean sojourn time at station i for lype r clients. given population1£
Nt •1Lk.]
mean number of clients at station i of type r . given population1£
fi,lkJ
A,.11J
Ii.,
c dIIICPU 1
IIICPU C
IIIDISK1 IIIDISK2Figure 2: A CPU-disk queuing network model
mean cycle time for clients of type r. given a system population
/£.:
a cycle consists of a predefined number of visits to each of the network stations mean throughput rate for clients of type r and given the system population/£.
mean workload for clients of type r at station i
mean number of visits per cycle to station i for a particular type r client CPU index. c
=
1 .2 ... Cindex disk unit. d = 1.2 .... D
We assume the processors to spread their attention equally over all clients in their queue: the processor sharing (PS) service discipline. In the mean value scheme this yields the fol-lowing relation for the sojourn time of a single visit to a CPU:
R
S, ...
lk]
= (
1+
r.
N,. ",I:&. -~] )w,. ,I 1 IThis relation is based on the arrival theorem for closed queuing networks. This theorem states that in product form queuing networks an client arriving at a certain queue will observe the equilibrium situation as if this arriving client observed the queue while not being a part of the system population. The arrival theorem was first mentioned in Laven-berg & Reiser [13].
The disk units are assumed to handle requests in first come first served (FCFS) order. The mean value relation we use to compute the sojourn times in the next sections. is the fol-lowing:
N
s" ..
l!..]
=
r.
Nd",I:&.
-£J. ]Wd ,,'+
wd,J (3),':= I
This relation. which is discussed e.g. in Bard [3]. yields exact solutions if the workloads are independent from the type of client. Note the similarity between this relation and the corresponding relation for the MIMll queue.
fi,I!J=
Elc"Sc"I!J+ Eld"Sd,,[k.]
(4)c d
The throughput of the system is defined as the average number of clients which finish a service cycle per unit of time. Noting that there are k, clients of type r in the system and that relation (4) yields in fact the average sojourn time for such a cycle of service, we can apply Little's formula ([14]) to the system as a whole. and thus nnd the system throughput:
rl.] kr
A,.
Js..
=
fir
[k.l k (5)The mean value scheme is completed by applying Little's formula to each of the separate queues in the network:
Ni.l [k.l
=
I
i.l A,I!.]si ,r[k.l (6)The relations (2) and (3) define the recursion in the mean value scheme, The recursion goes back until the zero population vector. at which level it is known that all queues are empty.
3. Aggregation
Now let us assume we consider a system as described in the introduction of this paper; a system in which each of the clients may visit only one of the CPU's. Furthermore. let us assume that in accordance with the workload characteristics we decide to aggregate a number of client types r to an aggregated type q. This could. but not necessarily. be a log-ical aggregation. e.g. aggregating all batch jobs or all compiler jobs. To denote this aggrega-tion. we introduce the following notation:
E>q Set of types r which are aggregated in type q
The workload characteristics of the aggregated type are largely determined by the relative throughput of the component parts. In case the system and workload characteristics are balanced. the aggregation is more or Jess trivial:
I
i,,,=
Wi,q=
(7) (8)'\"
Ii"
K, L" '\" Wi , lEe L. li",Ki ,,' , q ,'Ee
q (9)The problem which remains is of a reduced complexity and can be solved again with the mean value scheme (2)-(6).
The weighting of the workload characteristics (8)-(9) is less evident in case we observe an unbalanced system. For instance in the case we aggregate two types of clients dedicated to
different processors. whereas these processors show different performance characteristics. In the latter case. if we are interested in the number of jobs which finish their computer service, instead of the number of CPU or disk visits. we need to approximate the number of service cycles required; a job assigned to a CPU A may require more CPU visits in order to be completed than a similar job processed by a CPU B. For this purpose let us introduce the following notation:
v, The average number of cycles which is required for a job to finish service According to this definition a job of type r assigned to a CPU c requires on the average v,
f
C J' visits to the CPU ber ore being completed.The average number of cycles Vg required for an aggregated type of job to be completed is
determined by the v,'s of the composing types weighted with the corresponding throughputs. Approximating the relative throughput by the population ratios. we find the following relation:
Kg =
r.
K, v.,. ,.E 0 v,q
(10)
Here. as in Un-(9). the relative throughput is probably insufficiently reflected. Unfor-tunately. it is the throughput we are after with our calculations. which makes it hard for us to find a better aggregation then (8)-(10). This unless we are prepared to use an itera-tion to estimate the throughput. But then again. the use of iteraitera-tion increases the computa-tional complexity we tried to reduce.
As an alternative for the direct aggregation of the workload characteristics. (7)-(10). we suggest an adjustment of the mean value scheme. Again we aggregate the types of clients
r E 9q to a new type q. However in each of the recursion steps of the mean value scheme. we act as if we are dealing with the original R type of clients problem. Redefining
1£
as the aggregated population vector (k l.kz ... k" ... klJ). this leads to the following set of equa-tions: R ( 1+
.r.
Ni ;,[1£
-~] )Wj) .i a PS queues
1,1[1£]
= ,.': 1 R (11).r.
Ni"'Ik-~]Wi,,.,+
Wi" ,i a FCFS queueI"~ = 1
A,lkl
=
r.f
i " Si,,.lkl
(12) Ni , ,[1£]
=
f
i. . .
,Ai I IkJsi I[1£]
(13).r.
A,.Ik]/v, (14) ,.E 0qNote that if we aggregate two identical types of clients. both the aggregation (7)-(10) and the scheme (11 )-(14) will yield the exact results.
Both aggregation algorithms neglect the fact that the aggregation increases the variability of in particular the queue length distributions of the CPUs. An increasing variability usu-ally causes the system throughput to drop. This effect is best demonstrated in an example. Assume we have a small system consisting of two identical CPU's connected to a single disk unit. The system population consists of two identical clients. each assigned to a private CPU. We are interested in the throughput of the disk unit. According to the mean value scheme (2)-(6) we find. assuming the visit frequencies to be one:
01[(1.0)]
=
O2[(0.1)]=
Wd+
weand thus:
(15)
For both the aggregation schemes we find. however. that the CPU queues found by clients of the two client population are non empty:
O[(nJ
=
Wd+
We and thus: O.5wc O.5wc+
We+
W c -Wd+
We A[(2)]=
2 (16)It is clear that the approximative throughput expressed in equation (16) is strictly less than the exact throughput given in equation (15).
As stated before. the difference in the results of equation (15) and (16) is largely deter-mined by the increment in system variability. We may look at this problem in terms of the arrival theorem. The arrival theorem states that a client arriving at a station will
observe a system queue as if he was looking at the equilibrium of the system with one client of the arriving type removed from the population. A removal of a client from the system population has two effects on the observed equilibrium. The direct effect is that the queue lengths for the clients of the type from which one client is removed reduces: summing over all queues. we find one client less. A second. indirect effect is that some probably minor shifts in the lengths of the queues of other types may occur. but conserv-ing their total lengths. If we use the set of equations (11)-(14) and we study the example just treated. the removal of an aggregated client can be seen as the removal of half a CPU 1 client and half a CPU 2 client. Looking at the sojourn time at the disk unit we see that. as both types of clients are approximately equal. this removal of two times half a client has about the same effect on the total queue length at this station as the removal of a client of the arriving type: the arriving client observes two chains. each containing half a client less than they should contain according to the system population. At the CPU queue. however. we observe only half of the direct effect. as only one of the aggregated types visits this CPU. This will yield an overestimation of the queue length and therewith the sojourn time at the CPU. This observation leads to the following suggestion for an improved approximation of the sojourn time at stations at which only one type
r
E 9q arrives: (17) (18) where Kq _ Kg K ~ - (0.0 .... 0. K ,0 ... 0)'"
'"
a population vector containing type r clients only. As this population vector is likely to contain a non integer number of clients. an interpolation might be required. If we denote by
I
x1
the smallest integer equal to or larger than x. and byI
x1
the largest integer equal to or smaller than x. then this interpolation leads to the following expression for the average queue lengths:K
Ni.,l!.- K:
~]:=
o
(19)
.elsewhere
The use of equations (17)-(19) is based on the idea that the direct influence of the removal of a client is most important. whereas the indirect effect is almost independent from the
type of client removed. Note that with the use of this scheme, the results obtained for the two client example are exact.
A logical question to ask next, is why not use equations (17)-(19) for all the stations. The advantage would be that the total number of type r clients found in the queues is indeed one less than the number of type r clients present. However, we note that at the stations
at which other types r' of clients arrive which are part of the aggregated type q. the queues of types r' also feel a direct influence of the removal of an aggregated type. This would reduce these queue lengths. such that the beneficial effect of a more correct queue r
approximation would be override. Again. however. we may think of an artificial way to overcome these problems. by suggesting the following relations:
s"lkJ
= (
1+
Nc.r[t-~q
.£q] r (20) Sdl,[t]=
wd,r+
Nd.r[t-~: ~]Wd)'
+
,r.
Nd )"Ik
-~
]Wd)'+
,r.
k k"-1 Nd,I,1k
-~
]Wd)' (21) r9.eq , rESq q ,.' ;1f;.rThe effect of using these relations is that the sum of all queue lengths used to compute the sojourn times for a client of type r. yields exactly the total population considered in the recursion step minus this type r client.
In the next section we shall discuss some numerical examples to evaluate the four approxi-mation methods mentioned in this section.
4. Some numerical experiments
To allow a numerical evaluation of the aggregation methods discussed in the preceding sec-tion. we have created a test-bed of 64 examples. We distinguish between the examples according to several problem characteristics: the number of client types. the number of CPU·s. the bottleneck situation and the differences in the workload of the clients to be aggregated.
As to the first problem characteristic. we consider two categories of problems:
Ia} Only one type of client per CPU. After aggregating, a problem remains with only one type of client.
Ib) Two types of clients per CPU. In that case the aggregation defines two types of clients.
For the second problem characteristic. the number of CPU's in the model. we have two categories of examples. Note that if we want to study a model with more CPU's, this
automatically increases the level of aggregation: IIa) Two CPU-so
lIb) Three CPU·s.
The third problem characteristic determines the bottleneck station in the model. We con-sider the station with the largest utilization to be the bottleneck station. Four categories of problems are defined:
IlIa) The CPU is the bottleneck. The number of disk units is that large that there is no queuing at these devices. In the model the disk units are replaced by so called infinite-server stations.
IIIb) The CPU is the bottleneck. The average utilization of the disk units is about half the utilization of the bottleneck station.
IIIc) The CPU is the bottleneck. The average disk utilization is about 75% of the bottleneck station.
IIId) The disk units define the system bottleneck. The CPU utilization is about 75% of the utilization of this bottleneck.
In all examples the model parameters. are chosen such that the bottleneck station reaches an utilization between 75% and 90%.
The fourth problem characteristic is defined to study differences in the workloads of types of clients to be aggregated. These differences arise primarily if different CPU's are imple-mented in the system under study. Again. we distinguish between four categories of prob-lems:
IVa) All CPU's identical. No differences in workload.
IVb) All CPU's identical. Each CPU has a private system disk such that the disk workload of the types of clients to be distinguished differs.
IVc) The CPU's differ in processor speed by a factor 2 or 3. Each CPU has a private system disk. The types to be aggregated have approximately the same service requirements per visit. but differ in the number of visits required to finish their work.
IVd) Again the processors differ in speed by a factor 2 or 3. This is partly adjusted by reducing the multi programming level with a factor two or three as well. The number of visits before a job is finished is identical for each of the CPU·s. Each CPU has access to a private system disk again.
Note that for the problems resulting under IVa and IVb. the direct aggregation of the workload via (8)-(10) is identical to the mean value aggregation through (11)-(14).
The service requirements for a single disk unit were taken identical for all types of clients. thus al10wing the "exact" solution to be calculated.
For each combination of problem categories. we have defined one example. such that our test-bed comprises of 64 examples. As half of the examples defines two aggregated types of
clients. this leads to 96 observations of the aggregated turn-around-time. A set of tables is added at the end of this paper to give an overview of the obtained results.
In Tables 1 to 5 we summarized our major findings for the errors in the aggregated turn-around-time. The methods reviewed are:
1: Direct aggregation of workload: equation (8)-( 10)
2: Aggregation in the mean value scheme: equation (11)-(14) 3: Idem. but with adjustment (17)-(19)
4: Idem. but with adjustment (19)-(21)
In Table 1 the average errors found in comparing the exact and approximate aggregated turn-around-times are given. In Table 2 and 3 these errors are split into their positive and negative components. The direct workload aggregation always overestimates the system response time. as had been suggested in the preceding section. The same observation. some-what to our surprise. holds for the mean value scheme based approximations as well. The difference in the predictions through (17)-(19) and (19)-(21) are only very moderate. This is largely explained by the fact that the test-bed contained no examples in which the disk workload differed among the types of clients to be aggregated.
In Table 4 and 5 the error distribution is further analyzed by looking at the number of examples which gave less than 5% errors. and the maximum observed error. For the methods 3 and 4 these maximum figures are again only moderate.
Although with this report we intended to study a special type of aggregation. aggregation over the processors. we have also performed a small exercise in aggregating the clients per processor instead. The results of this exercise. which comprises of the 32 examples conven-ing under problem characteristic lb. are summarized in Table 6. The results suggest that this type of aggregation is much more robust. Especially aggregation method 2 behaves very well under these conditions. Note that. as each of the composing types in the aggrega-tion have the same routing pattern. aggregaaggrega-tion methods 2 and 3 are identical.
S. The computer-terminal model
All the examples discussed in the preceding section dealt with batch processing systems. Let us now suppose that one of the aggregated types of clients is fed as a terminal stream. We may analyze this system. basing ourselves on the ideas of the parametric analysis.
[S.6], as a computer-terminal model. In this model. see e.g. Figure 3. the computer is seen
as a single station with a queue dependent service rate. These service rates can be obtained by solving the underlying centra] server model.
The system may possess an external queuing mechanism; the computer may allow less jobs to enter and being processed simultaneously then there are maximally offered. The max-imum number of jobs allowed in the system is known as the multi programming level. Usually. for each of the processors such a level is defined. Jobs which are offered when the system uses its full multi programming level. are temporary stored in the "memory queue". The underlying central server model has to be solved for all populations which do
not exceed these multi programming levels.
The workloads of the jobs offered to the computer system are assumed to be exponentially distributed. This model usually performs very well. see e.g. Salza & Lavenberg (16] for a discussion. and we shall assume that it yields the exact results.
T E R
r.r
N~
T EIl
I N~
Figure 3: A double queue computer-terminal model A computer structure such as depicted in Figure la suggests a single queue system. Once a user has "logged on" to the system. its identity is administrated in the main memory and therewith known to both the processors. Therefore. each job offered to the system can be assigned to an arbitrary processor. It is the task of the "system scheduler" to provide both the processors a fair share of the offered load. We a:>sume that a job offered to the system is always assigned to the proce~'"Sor which has the lowest utilization of its multi program-ming level as observed at the moment the job is allowed to enter the system. Once a job has been assigned. it is not a))owed to change from processor until it is finished.In Tables 7 to 11 we have summarized our major findings for the analysis of this single queue system. Again. we used the predicted response times to compare the approximations with the "exact" results. Apart from a minor modification of the multi programming lev-els. we used the same test-bed of examples to represent the computer workload as in the preceding section. Por each of the processors. the maximum multi programming level for jobs defined by the interactive system users was chosen one above the multi programming level as used in the preceding section. The number of terminals connected to the system was chosen twice as high as this maximum multi programming level. The think times were taken approximately equal to the response times in the originating central server model problem. In this way we have provided that the test-bed possesses approximately the same utilization characteristics as in the preceding section.
The exact solution to the computer terminal problem was based on the iterative solution method for solving Markov chains as discussed in Van der Wal & Schweitzer [20).
In
Table 7 the average response time errors are given. The mean value aggregations 3 and 4 prove to perform just as well as under the assumptions made in the preceding section. Approximation method 1 and 2 perform even worse than they did before.As one would guess after looking at the results of Table 2 and 3. most of the observed errors are positive. The positive and the negative components of the errors are given in
Table 8 and 9.
In Table 10 and 11. the errors are further analyzed. The maximum error observed for aggregation method 3 and 4 are moderate: less than 5%. i'he errors of the first and second aggregation method tend to raise to an unacceptable high level.
The system depicted in Figure 1 b suggests a double queue system, like the one depicted in Figure 3. An user may log on to any of the processors, but once logged on all the jobs it offers to the system are automatically assigned to the processor at which the user identity is known.
In Table 12 to 15 we have compared the results of the aggregated problem with the results of the problem in which each of the CPU's indeed possesses its own memory queue. In case of the aggregated problem, we nevertheless treat the system as the single queue system as described above. We used the same test-bed of examples as used to obtain Table 7 to 11. The aggregation results. however. are slightly worse in this case. The errors which appear in this type of problem are a bit like when comparing the answers of the average response times in two MIMII queues with the response times in one composite MIMl2 system.
The overall average error for each of the aggregation methods is approximately the same; a 7% error level. In contrast with the preceding examples, we see that there is a major cause for this high error level. at least as far as we are concerned with the aggregation methods 3
and 4. If we study a system with identical processors, the errors are very acceptable. If we study a system with different processors, the errors tend to be high. Apparently. in the single queue situation the fast processors can be used more intensively then in the muhi queue situation. This causes an overestimation for the predicted response time; the aggre-gated solution is based on the single queue solution.
In Table 13 and 14 the errors are split into their positive and negative components again. We note that the positive component is largely due to the predictions for the terminal type of clients. whereas the negative component is due to the approximations for the batch type of clients.
In Table 15 the maximum observed errors are presented. Note again that under problem characteristic IVa and IVb. the aggregation methods 3 and 4 attained a very acceptable error leveL
6. Conclusions
We have discussed and numerically evaluated an aggregation method based on the mean value scheme. We have shown that in aggregating types of clients which partly visit different stations. the direct workload aggregation is insufficient. The aggregation in the mean value scheme suggested in this paper. can solve this problem with just a small loss in computational efficiency as compared to the direct workload aggregation.
Three variants of the mean value scheme based aggregation method were discussed. The first method just differ.ed from the ordinary mean value scheme in the recursion used. In many situations it can be proved to be identical to the direct workload aggregation. This method proved to be especially accurate in the situation where the types of clients to be aggregated have the same routing pattern.
differed.
The aggregation algorithms are especially interesting, if the problem to be solved is to be used in the parametric analysis of a larger system. Although the starting point of this model was the central server model and the parametric analysis of a computer terminal problem. the aggregation methods suggested may be applied to tackle other large problems as well.
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21. WIJBRANDS. R.J .. "The Analysis of 1/0 Configurations: On the Robustness of the Central Server Model:' in System Modelling and Optimization: Proceedings of the 12th IFIP Conference. ed. A. Prekopa. J. Szelezsim. and B. Strazicky. Springer Verlag.
IA) 7.15 6.68 0.80 0.80
m)
7.79 7.21 1.62 1.61 IIA) 5.67 5.38 1.12 1.13 lIB) 9.48 8.69 1.57 1.56 rnA) 9.02 8.50 1.16 1.16 IlIB) 10.63 9.60 1.67 1.78 me) 7.47 6.99 1.48 1.55 IIID) 3.19 3.04 1.07 0.88 IVA) 7.03 7.03 1.39 1.42 IVB) 6.92 6.92 1.35 1.33 Ive) 7.67 7.26 1.33 1.32 IVD) 8.69 6.92 1.32 1.30 ALL 7.58 7.03 1.35 1.34Table 1: Central server model with aggregation over the processors: Average response time errors (%)
EXAMPLE IMETHOO 1 2 3 4 IA) 7.15 6.68 0.80 0.80
m)
7.79 7.21 1.62 1.61 IIA) 5.67 5.38 1.12 1.13 liB) 9.48 8.69 1.57 1.56 IlIA) 9.02 8.50 1.16 1.16 IlIB) 10.63 9.60 1.67 1.78 I1Ie) 7.47 6.99 1.48 1.55 IIID) 3.19 3.04 1.07 0.88 IVA) 7.03 7.03 1.39 1.42 IVB) 6.92 6.92 1.35 1.33 IVe) 7.67 7.26 1.33 1.32 IVO) 8.69 6.92 1.32 1.30 ALL 7.58 7.03 1.35 1.34Table
2:
Central server model with aggregation over the processors: Average positive component of the respalse t~me error (%)EXAMPLE IMETHOO 1 2 3 4 IA) 0.00 0.00 0.00 0.00
m)
0.00 0.00 0.00 0.00 IIA) 0.00 0.00 0.00 0.00 IIB) 0.00 0.00 0.00 0.00 IlIA) 0.00 0.00 0.00 0.00 I1IB) 0.00 0.00 0.00 0.00 me) 0.00 0.00 0.00 0.00 IlIO) 0.00 0.00 0.00 0.05 IVA) 0.00 0.00 0.00 0.00 IVB) 0.00 0.00 0.00 0.00 Ive) 0.00 0.00 0.00 0.00 IVO) 0.00 0.00 0.00 0.00 ALL 0.00 0.00 0.00 0.00Table 3: Central server model with aggregation over the processors:
IA) 31.25 31.25 100.00 100.00 IB) 23.44 25.00 100.00 100.00 IIA) 35.42 37.50 100.00 100.00 lIB) 16.67 16.67 100.00 100.00 IlIA) 4.17 4.17 100.00 100.00 mB) 0.00 0.00 100.00 100.00 me) 16.67 20.83 100.00 100.00 llID) 83.33 83.33 100.00 100.00 NA) 33.33 33.33 100.00 100.00 NB) 29.17 29.17 100.00 100.00 Ive) 20.83 20.83 100.00 100.00 ND) 20.83 25.00 100.00 100.00 ALL 26.04 27.08 100.00 100.00
Table 4: Central server model with aggregation over the processors: Percentage of observed respLse times with a 5% or less error
EXAMPLE
I
METHOD 1 2 3 4 IA) 13.80 13.05 1.57 1.74 IB) 20.45 14.96 2.87 2.95 IIA) 11.15 9.79 2.40 2.43 lIB) 20.45 14.96 2.87 2.95 lIlA) 14.96 14.96 2.74 2.74 mB) 20.45 13.90 2.87 2.95 me) 14.72 11.79 2.64 2.73 IIID) 7.55 6.96 1.62 1.47 IVA) 14.96 14.96 2.87 2.95 IVB) 13.79 13.79 2.69 2.76 Ive) 15.30 13.90 2.68 2.73 IVD) 20.45 13.32 2.53 2.60 ALL 20.45 14.96 2.87 2.95Table 5: Central server model with aggregation over the processors: M.aximum response time error observed (%)
EXAMPLE IMETIiOD 1 2 3 4 IIA) 1.33 0.14 0.14 0.70 nB) 1.04 0.04 0.04 0.40 lIlA) 1.14 0.16 0.16 1.02 IIIB) 0.93 0.09 0.09 0.73 me) 1.99 0.05 0.05 0.27 IIID) 0.57 0.04 0.04 0.05 NA) 1.33 0.13 0.13 0.55 IVB) 1.19 0.06 0.06 0.54 Ive) 0.71 0.07 0.07 0.47 ND) 1.40 0.07 0.07 0.52 ALL 1.16 0.08 0.08 0.52
Table 6: Central server malel with aggregation per proassor: Average response time errors (%)
IA) 11.80 10.02 1.00 1.00 IB) 9.83 8.39 1.81 1.79 IlA) 7.97 7.10 1.27 1.27 lIB) 13.00 10.77 1.81 1.79 lIlA) 14.02 12.04 1.63 1.63 IIlB) 15.12 12.40 1.64 1.79 me) 9.77 8.49 1.69 1.80 IIID) 3.02 2.81 1.19 0.89 IVA) 9.18 9.18 1.66 1.70 IVB) 9.00 9.00 1.58 1.55 Ive) 9.31 8.70 1.36 1.36 IVD) 14.44 8.86 1.55 1.51 ALL 10.48 8.93 1.54 1.53
Table
7:
ParCUTlStric analysis of a single queue computer-terminol model: average response time errors (%)EXAMPLE / METIIOD 1 2 3 4 IA) 11.80 10.02 0.81 0.81 IB) 9.83 8.39 1.74 1.72 IIA) 7.97 7.10 1.21 1.21 UB) 13.00 10.77 1.65 1.62 IlIA) 14.02 12.04 1.31 1.31 IIlB) 15.12 12.40 1.61 1.77 me) 9.77 8.49 1.65 1.75 I1ID) 3.02 2.81 1.16 0.84 IVA) 9.18 9.18 1.62 1.66 IVB) 9.00 9.00 1.55 1.52 IVC) 9.31 8.70 1.08 1.07 IVD) 14.44 8.86 1.47 1.43 ALL 10.48 8.93 1.43 1.42
Table 8: ParCUTlStric analysis of a single queue computer-terminol model Average positive component of the response time errcr (%)
EXAMPLE / METIlOD 1 2 3 4 IA) 0.00 0.00 -0.19 -0.19
m)
0.00 0.00 -0.07 -0.07 UA) 0.00 0.00 -0.05 -0.05 liB) 0.00 0.00 -0.16 -0.17 mA) 0.00 0.00 -0.33 -0.33 mB) 0.00 0.00 -0.03 -0.02 me) 0.00 0.00 -0.05 -0.05 I1ID) 0.00 0.00 -0.03 -0.05 IVA) 0.00 0.00 -O.D4 -0.04 IVB) 0.00 0.00 -0.03 -0.03 IVC) 0.00 0.00 -0.28 -0.29 IVD) 0.00 0.00 -0.08 -0.08 ALL 0.00 0.00 -0.11 -0.11Table 9: ParCUTlStric analysis of a single queue computer-terminol model: Average negative component of the response time error (%)
IA) 25.00 25.00 100.00 100.00
m)
25.00 26.56 100.00 100.00 llA) 29.17 29.17 100.00 100.00 llB) 20.83 22.92 100.00 100.00 IIlA) 0.00 0.00 100.00 100.00 IIIB) 0.00 0.00 100.00 100.00 mC) 8.33 12.50 100.00 100.00 IlID) 91.67 91.67 100.00 100.00 IVA) 29.17 29.17 100.00 100.00 IVB) 29.17 29.17 100.00 100.00 IVC) 20.83 20.83 100.00 100.00 IVD) 20.83 25.00 100.00 100.00 ALL 25.00 26.04 100.00 100.00 iTable 10: Parametric analysis of a single queue computer-terminal model: Percentage of observed resJK1l1.se times with a 5% or less error
EXAMPLE
I
METHOD 1 2 3 4 IA) 32.48 19.19 2.26 2.52m)
29.64 18.77 4.37 4.43 llA) 15.90 13.54 3.33 3.35 lIB) 32.48 19.19 4.37 4.43 IlIA) 32.48 18.90 4.07 4.07 IIIB) 29.64 19.19 4.37 4.43 IIIC) 18.78 13.34 3.56 3.65 IIID) 7.87 7.21 2.35 2.16 IVA) 18.90 18.90 4.07 4.08 IVB) 19.19 19.19 3.77 3.84 IVC) 20.18 17.35 4.36 4.41 IVD) 32.48 16.25 4.37 4.43 ALL 32.48 19.19 4.37 4.43Table 11: Parametric analysis of a single queue computer-terminal model: Maximum res]XYfI.setime error observed (%)
EXAMPLE
I
METHOD 1 2 3 4 IA) 4.79 5.33 6.58 6.59 m) 8.77 8.29 5.83 5.85 IIA) 5.37 5.43 4.73 4.74 1m) 9.51 9.18 7.43 7.45 IlIA) 8.55 8.69 7.45 7.45 IIIB) 10.04 9.22 6.90 6.78 IIIC) 7.24 7.33 6.23 6.16 IIID) 3.92 3.98 3.75 4.00 IVA) 7.34 7.34 2.86 2.86 IVB) 7.07 7.07 2.87 2.90 IVC) 6.85 6.67 8.36 8.38 IVD) 8.51 8.14 10.23 10.27 ALL 7.44 7.30 6.08 6.10Table 12: Parametric analisis of multiple queue computer-terminal model: Average res]XYfI.se time errar (%)
IA) 4.14 3.68 0.00 0.00 IB) 8.00 7.08 2.58 2.58 IIA) 4.95 4.58 1.39 1.39 lIB) 8.48 7.31 2.05 2.05 IlIA) 8.08 7.28 1.48 1.48 IlIB) 10.00 8.34 2.05 2.07 IlIC) 6.37 5.81 1.95 1.97 IIID) 2.41 2.34 1.39 1.36 IVA) 7.19 7.19 1.24 1.26 IVB) 6.98 6.98 1.24 1.23 IVC) 5.06 4.68 1.83 1.83 IVD) 7.63 4.93 2.56 2.56 ALL 6.71 5.94 1.72 1.72
Table 13:Parametric analysis of a multiple queue computer-terminol model Average positive component of response time error (%)
EXAMPLE
I
METHOD 1 2 3 4 IA) -0.64 -1.66 -6.58 -6.59 IB) -0.77 -1.21 -3.25 -3.27 IIA) -0.42 -0.85 -3.34 -3.35 lIB) -1.03 -1.87 -5.38 -5.41 lIlA) -0.47 -1.41 -5.97 -5.97 IIIB) -0.04 -0.87 -4.85 -4.71 IIIC) -0.87 -1.52 -4.28 -4.19 IIID) -1.51 -1.64 -2.35 -2.64 IVA) -0.15 -0.15 -1.62 -1.60 IVB) -0.09 -0.09 -1.64 -1.67 IVC) -1.79 -1.99 -6.53 -6.55 IVD) -0.87 -3.21 -7.67 -7.71 ALL -0.73 -1.36 -4.36 -4.38Table 14:Parametric analysis of a multiple queue computer terminol model Average negative component in response time error (%)
EXAMPLE
I
METHOD 1 2 3 4 IA) 14.00 14.00 22.18 22.18 IB) 37.13 22.97 14.89 14.89 IIA) 22.11 18.03 15.70 15.70 lIB) 37.13 22.97 22.18 22.18 IlIA) 25.18 20.07 22.18 22.18 IlIB) 37.13 22.97 18.86 18.63 IlIC) 24.55 18.03 14.41 14.21 IIID) 12.13 11.97 11.62 11.90 IVA) 20.07 20.07 5.73 5.80 IVB) 18.50 18.50 5.62 5.69 IVC) 24.05 21.91 15.85 15.85 IVD) 37.13 22.97 22.18 22.18 ALL 37.13 22.97 22.18 22.18Table 15:Parametric analysis of a multiple queue computer-terminol model Maximum response time error observed (%)
